the churches are all free, so hit those when you run into them until you get bored. def. go to st peters even if you have to stand in line (but try not to stand in line too long). go to the top of it while you're there. there is a small 3rd or 4th century church in the suburbs that i went to once that was cool. i can find the name and location if you're interested. the pantheon is an incredible building. you can sneak into the colesseum at night and have it to yourself, so do that. the forum is right there, so you'll probably want to hit it, even though it's kinda overrated (cool stuff, but too crowded to enjoy). the baths of some emperor were pretty cool if you're down in that part of town. i liked walking along the via appia in the area outside the walls where there's a bunch of aqueduct arches.
a lot of times, you can just kinda wander around seeing stuff and it's mostly all old. went somewhere once that was like a museum and behind it was a garden with old marble sculptures and domestic rabbits hopping around the garden. didn't go in the museum, but the garden was enjoyable. hopped a wall and walked around a monastery or something with fruit trees and stray cats. also nice. just stuff like that is fun. don't go when it's hot.
try to avoid the bad pizza and to eat a shitload of the good.